Distribution of Zeckendorf expressions

TL;DR

This paper studies the distribution of Zeckendorf representations, providing asymptotic formulas for the count of such representations below a threshold, and extends the analysis to general linear recurrences.

Contribution

It derives asymptotic formulas for the number of Zeckendorf-like representations and generalizes these results to broader linear recurrence settings.

Findings

Asymptotic formulas for Zeckendorf representations

Extension to general linear recurrences

Insights into the distribution of non-adjacent Fibonacci sums

Abstract

By Zeckendorf's Theorem, every positive integer is uniquely written as a sum of distinct non-adjacent Fibonacci terms. In this paper, we investigate the asymptotic formula of the number of binary expansions $<x$ that have no adjacent terms, and generalize the result to the setting of general linear recurrences with non-negative integer coefficients.